Determination of uniformly most powerful tests in discrete sample spaces |
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| Authors: | Hagen Scherb |
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| Institution: | (1) Institute of Biomathematics and Biometry, GSF-National Research Center for Environment and Health, D-85764 Neuherberg, Germany (e-mail: scherb@gsf.de), DE |
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| Abstract: | Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H
0 and the alternative hypothesis H
1 are given by
(i) H
0 : θ≤θ0, H
1 : θ>θ0, and
(ii) H
0 : θ≤θ1 or θ≥θ2, H
1 : θ1<θ<θ2, where θ1<θ2.
Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H
0 and H
1 take the form
(iii) H
0 : θ1≤θ≤θ2, H
1 : θ<θ1 or θ>θ2, where θ1<θ2, and
(iv) H
0 : θ=θ0, H
1:θ≠θ0.
To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values
c
1, c
2 and two randomization constants γ1, γ2. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases
is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete
sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination
of UMP tests in cases (ii) through (iv).
Received: March 1999 |
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| Keywords: | : Discrete sample space Uniformly most powerful test Determination of critical values Algorithm |
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